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Theorem fvopab4ndm 6898
Description: Value of a function given by an ordered-pair class abstraction, outside of its domain. (Contributed by NM, 28-Mar-2008.)
Hypothesis
Ref Expression
fvopab4ndm.1 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
Assertion
Ref Expression
fvopab4ndm 𝐵𝐴 → (𝐹𝐵) = ∅)
Distinct variable group:   𝑥,𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem fvopab4ndm
StepHypRef Expression
1 fvopab4ndm.1 . . . . 5 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
21dmeqi 5810 . . . 4 dom 𝐹 = dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
3 dmopabss 5824 . . . 4 dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ⊆ 𝐴
42, 3eqsstri 3959 . . 3 dom 𝐹𝐴
54sseli 3921 . 2 (𝐵 ∈ dom 𝐹𝐵𝐴)
6 ndmfv 6798 . 2 𝐵 ∈ dom 𝐹 → (𝐹𝐵) = ∅)
75, 6nsyl5 159 1 𝐵𝐴 → (𝐹𝐵) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1541  wcel 2109  c0 4261  {copab 5140  dom cdm 5588  cfv 6430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pr 5355
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3432  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-br 5079  df-opab 5141  df-dm 5598  df-iota 6388  df-fv 6438
This theorem is referenced by:  fvmptndm  6899
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